Rational points on $X_0^+ (N)$ and quadratic $\mathbb {Q}$-curves

Galbraith, Steven D.

Rational points on

X_{0}^{+} (N)

and quadratic

ℚ

-curves

Galbraith, Steven D.

Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 1, pp. 205-219.

Résumé
Abstract

Nous considérons les points rationnels sur $X_{0} (N) / W_{N}$ dans le cas où $N$ est un nombre composé. Nous faisons une étude de certains cas qui ne se déduisent pas des résultats de Momose. Des points rationnels sont obtenus pour $N = 91$ et $N = 125$ . Nous exhibons aussi les $j$ -invariants des $ℚ$ -courbes quadratiques correspondantes.

The rational points on $X_{0} (N) / W_{N}$ in the case where $N$ is a composite number are considered. A computational study of some of the cases not covered by the results of Momose is given. Exceptional rational points are found in the cases $N = 91$ and $N = 125$ and the $j$ -invariants of the corresponding quadratic $ℚ$ -curves are exhibited.

MR Zbl | 2 citations dans Numdam

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     title = {Rational points on $X_0^+ (N)$ and quadratic $\mathbb {Q}$-curves},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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     publisher = {Universit\'e Bordeaux I},
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Galbraith, Steven D. Rational points on $X_0^+ (N)$ and quadratic $\mathbb {Q}$-curves. Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 1, pp. 205-219. http://www.numdam.org/item/JTNB_2002__14_1_205_0/

Bibliographie
Cité par

[1] A.O.L. Atkin, J. Lehner, Hecke Operators on Γ0(N), Math. Ann. 185 (1970), 134-160. | Zbl

[2] B.J. Birch, Heegner points of elliptic curves, AMS Symp. math. 15 (1975), Inf. teor., Strutt. Corpi algebr., Convegni 1973, 441-445. | MR | Zbl

[3] H. Cohen, N.-P. Skoruppa, D. Zagier, Tables of modular forms. Preprint, 1992.

[4] J.E. Cremona, Algorithms for modular elliptic curves. Cambridge (1992) | MR | Zbl

[5] P. Deligne, M. Rappoport, Les schemas de modules de courbes elliptiques. In Modular Functions one Variable II, Springer Lecture Notes Math. 349 (1973), 143-316. | MR | Zbl

[6] N. Elkies, Remarks on elliptic K-curves, preprint, 1993.

[7] N. Elkies, Elliptic and modular curves over finite fields and related computational issues. In D. A. Buell and J. T. Teitelbaum (eds.), Computational Perspectives on Number Theory, AMS Studies in Advanced Math., 1998, 21-76. | MR | Zbl

[8] S.D. Galbraith, Equations for Modular Curves. Doctoral Thesis, Oxford, 1996.

[9] S.D. Galbraith, Rational points on X0+(p). Experiment. Math. 8 (1999), 311-318. | MR | Zbl

[10] S.D. Galbraith, Constructing isogenies between elliptic curves over finite fields. London Math. Soc. J. Comp. Math. 2 (1999), 118-138. | MR | Zbl

[11] J. González, Equations of hyperelliptic modular curves. Ann. Inst. Fourier bf41 (1991), 779-795. | Numdam | MR | Zbl

[12] J. González, J.-C. Lario, Rational and elliptic parametrizations of Q-curves. J. Number Theory 72 (1998), 13-31. | MR | Zbl

[13] J. González, On the j-invariants of the quadratic Q-curves. J. London Math. Soc. 63 (2001), 52-68. | MR | Zbl

[14] B.H. Gross, Arithmetic on elliptic curves with complex multiplication. Lect. Notes Mathematics 776, Springer, 1980. | MR | Zbl

[15] B.H. Gross, Heegner Points on X0(N). In Modular Forms, R. A. Rankin (ed.), Wiley, 1984, 87-105. | MR | Zbl

[16] B.H. Gross, D.B. Zagier, On singular moduli. J. Reine Angew. Math. 355 (1985), 191-220. | MR | Zbl

[17] Y. Hasegawa, Table of quotient curves of modular curves X0(N) with genus 2. Proc. Japan Acad. Ser. A 71 (1995), 235-239. | MR | Zbl

[18] Y. Hasegawa, Q-curves over quadratic fields. Manuscripta Math. 94 (1997), 347-364. | MR | Zbl

[19] M.A. Kenku, On the Modular Curves X0(125), X0(25) and X0(49). J. London Math. Soc. 23 (1981), 415-427. | MR | Zbl

[20] S. Lang, Elliptic Functions, 2nd edition. Springer GTM 112, 1987. | MR | Zbl

[21] B. Mazur, Modular Curves and the Eisenstein Ideal. Pub. I.H.E.S, 47 (1977), 33-186. | Numdam | MR | Zbl

[22] F. Momose, Rational Points on X0+(p r). J. Faculty of Science University of Tokyo Section 1A Mathematics 33 (1986), 441-466. | MR | Zbl

[23] F. Momose, Rational Points on the Modular Curves X +0(N). J. Math. Soc. Japan 39 (1987), 269-285. | MR | Zbl

[24] N. Murabayashi, On normal forms of modular curves of genus 2. Osaka J. Math. 29 (1992), 405-418. | MR | Zbl

[25] A. Ogg, Rational Points on Certain Elliptic Modular Curves. In H. Diamond (ed.), AMS Proc. Symp. Pure Math. 24, 1973, 221-231. | MR | Zbl

[26] A. Ogg, Hyperelliptic Modular Curves. Bull. Soc. Math. France 102 (1974), 449-462. | Numdam | Zbl

[27] K. Ribet, Abelian varieties over Q and modular forms. Proceedings of KAIST workshop (1992), 53-79.

[28] M. Shimura, Defining equations of modular curves X0(N). Tokyo J. Math. 18 (1995), 443-456. | Zbl